Choice of null-models

𝑘=1

Observations below the null-model prediction show shorter nearest neighbour distances, and thus, clumping. Similar to the pair-correlation function, the nearest neighbour distribution function is point-based, i.e. it describes spatial correlation between individuals. Here, we summarize the function by reporting the distance at which 90% of the trees have their nearest neighbour.

Empty space function (spherical contact distribution function)

The empty-space function is location-based, i.e. it describes the spatial distribution of individuals relative to random locations, not between individuals. The spatial distribution is described in terms of a frequency distribution of distances between random locations and the nearest tree. If the observed pattern is below the null-model prediction, the observed empty space is smaller than expected. This indicates regularity. Observations above the predicted values show large empty spaces and thus, indicate clustering in presence of also short empty-space distances. Please note that for Poisson processes nearest neighbour and empty-space functions are the same function (Stoyan et al. 1995).

Thus, if no deviations from the simulated Poisson process are observable, empty space and nearest neighbour distribution look the same. Here, we summarize the empty-space function by reporting the distance for which for 90% of random locations there is at least one tree found at or within this distance.

1.2.4 Choice of null-models

Null-Models are usually used to produce simulated randomised reference patterns of spatial characteristics that are anticipated in the absence of specific ecological processes (Gotelli and Graves 1996). Rejection of a null-model then indicates the presence of the

24 respective ecological processes. Taking the opposite perspective, null-models can also be seen as representing a particular spatial process such as Complete Spatial Randomness (CSR) or clustering. Then, the non-rejection of a null-model indicates that the process represented by the null-model could be the main driver of the observed pattern (Wiegand and Moloney 2004). Here, we thus consider as the best null-model the one that produces simulation envelopes with the smallest deviations in respect to the observed pattern, for an example graphic, see Figure 1-2.

Figure 1-2 Example (Ash-Ash) result of a pair-correlation function and presentation as quantum plots. The results of the pair-correlations will be presented as the lower coloured plot (quantum plot) only. Deviations from the simulation envelope towards clustering are green, deviations towards repulsion are red. If the function follows the envelope, the quantum plot is grey. Therefore, the null-model that shows where the pair-correlation function shows little deviation from the envelope will have a mostly grey simulation envelope.

Complete Spatial Randomness (CSR)

The above mentioned CSR is the simplest null-model. CSR assumes the absence of all spatial pattern-generating ecological processes that could lead to anisotropy or non-stationarity (Baddeley et al. 2015 p. 409). It corresponds to a homogeneous Poisson process (Cressie 1993 p. 586). In our particular study, the assumption of CSR implies that all locations have the same suitability for tree occurrence and thus the same probability

25 of holding trees (Specific Hypothesis 1, short SH1, null-model: CSR (nullubiq) with homogeneous pair-correlation function gubiq).

Including heterogeneity

However, heterogeneity was observed for soil properties (Mund 2004). Thus, we investigated if this heterogeneity also showed in the spatial tree patterns (technically, by rejecting SH1) and then separately applied two null-models to include two different forms of habitat heterogeneity: The first assumed species do not differ in their reaction to abiotic heterogeneity (Aim 2), the second allowed for differences depending on species identity (Aim 3). In lack of an explicit habitat model that included influential abiotic field-measured variables, we characterized habitat heterogeneity by tree density, considering tree density as the indirect outcome of differences in habitat suitability (Baddeley et al.

2000). To support the null-model choice and deal with the difficulties of estimating first and second order properties from the same model, we included ecological pre-knowledge (Diggle and Ribeiro Jr 2007). The deliberate inclusion of explicit pre-knowledge also promotes the possibility of separating between clumping due to heterogeneity, due to niche properties or due to tree-tree interactions. We assumed that large trees (> 30 cm DBH) are mature trees and that their differences in density indicated differences in abiotic habitat suitability. These trees had already survived the thinning process caused by adverse abiotic conditions. Following the approach of Getzin et al. (2008), we first hypothesized that suitability is equal for all species (Specific Hypothesis 2, short SH2, null-model: heterogeneous Poisson process (nullequal) with inhomogeneous pair-correlation function gequal).

However, depending on species identity, abiotic habitat heterogeneity may have a different impact on occurrence probability. Indeed, Zhang et al. (2013) observed that niche effects seem to be more important than stochastic processes in a temperate forest.

Thus, as an alternative to assuming equal habitat suitability for all species, we secondly hypothesized species-specific suitability. To this end, we characterised species-specific experienced habitat heterogeneity based on tree density of large trees (> 30 cm DBH) of each species individually (Specific Hypothesis 3, short SH3, null-model: inhomogeneous Poisson process (nullniche) with inhomogeneous pair-correlation function gniche).

Using a non-parametric approach, we estimated the intensity λ(x) of the spatial distribution of mature trees overall and species-wise for the two null-models SH2 and

26 SH3, respectively, by applying a moving window approach combined with an Epanechnikov kernel as suggested in Stoyan and Stoyan (1994b, 2008) and Getzin et al.

(2008). Following Baddeley et al. (2000), technically, the intensities λ(x) are not part of the null-models but are used as thinning surfaces to adjust the pair-correlation function (gequal or gniche).